Glossary of algebraic groups

There are a number of mathematical notions to study and classify algebraic groups.

In the sequel, G denotes an algebraic group over a field k.

notion explanation example remarks
linear algebraic group A Zariski closed subgroup of for some n Every affine algebraic group is isomorphic to a linear algebraic group, and vice versa
affine algebraic group An algebraic group which is an affine variety , non-example: elliptic curve The notion of affine algebraic group stresses the independence from any embedding in
commutative The underlying (abstract) group is abelian. (the additive group), (the multiplicative group),[1] any complete algebraic group (see abelian variety)
diagonalizable group A closed subgroup of , the group of diagonal matrices (of size n-by-n)
simple algebraic group A connected group which has no non-trivial connected normal subgroups
semisimple group An affine algebraic group with trivial radical , In characteristic zero, the Lie algebra of a semisimple group is a semisimple Lie algebra
reductive group An affine algebraic group with trivial unipotent radical Any finite group, Any semisimple group is reductive
unipotent group An affine algebraic group such that all elements are unipotent The group of upper-triangular n-by-n matrices with all diagonal entries equal to 1 Any unipotent group is nilpotent
torus A group that becomes isomorphic to when passing to the algebraic closure of k. G is said to be split by some bigger field k' , if G becomes isomorphic to Gmn as an algebraic group over k'.
character group X(G) The group of characters, i.e., group homomorphisms
Lie algebra Lie(G) The tangent space of G at the unit element. ) is the space of all n-by-n matrices Equivalently, the space of all left-invariant derivations.

Notes

  1. These two are the only connected one-dimensional linear groups, Springer 1998,Theorem 3.4.9

References

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